Is there a name that would be understandable to an undergraduate student who hasn't read anything about semigroups but has had a first course in algebra and knows what a semigroup/monoid is? What name would it be good to go under in a list of order-three semigroups?

@Asaf I haven't been so embarrassed for quite some time. Mocking may lead to an attempt at burying my head in the floor. That's probably safer than using sand, but still not a good idea. :)
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user23211Jul 2 '12 at 22:18

@ymar, if you prefer we can give it a Polish name, Stanislaw, perhaps.
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Asaf KaragilaJul 2 '12 at 22:26

Thanks. So if $G$ is a group, then $G^0$ ($G$ with zero adjoined) is called "$G$ with errors"?
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user23211Jul 2 '12 at 22:24

1

@ymar Not zero, but an element that spreads like a virus (usually denoted by $\bot$). Any operation can result in an error $\bot$ (e.g. if you divide $1$ by $0$ in some algebra containing division) and if the "error result" is passed to any other expression, it propagates, i.e. the result of that expression has to be $\bot$ as well. Indeed, that kind of structures are denoted $G^\bot$, and called "$G$ with errors" or "$G$ with error handling". This is very similar to NaNs or monad Maybe from Haskell language (e.g. $G^\bot \cong \mathtt{Maybe}\ G$).
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dtldarekJul 3 '12 at 6:09